To write a linear equation relating the height \( h \) to the length of the radius \( r \), we'll use the slope-intercept form of a linear equation, which is:
\[ h = mr + b \]
Here, \( m \) is the slope of the line, which represents how much \( h \) (height) changes for each single unit of change in \( r \) (radius length), and \( b \) is the y-intercept, which is the value of \( h \) when \( r = 0 \).
Given that height increases by 3.9 inches for each 1-inch increase in the length of the radius, our slope \( m \) will be:
\[ m = 3.9 \]
We are also given a specific point on the line that corresponds to a 70-inch-tall woman with an 8-inch-long radius. This gives us a point \( (r, h) = (8, 70) \).
Using this point, we can solve for the y-intercept \( b \) by plugging \( r \) and \( h \) into the slope-intercept equation:
\[ 70 = 3.9(8) + b \]
Now, we'll solve for \( b \):
\[ 70 = 31.2 + b \]
\[ 70 - 31.2 = b \]
\[ b = 38.8 \]
So the y-intercept of our linear equation is 38.8. Now we can write our final linear equation:
\[ h = 3.9r + 38.8 \]
This equation can be used to calculate the height \( h \) of a woman based on the length \( r \) of her radius bone.